Theorem termchomn0

Description: All hom-sets of a terminal category are non-empty. (Contributed by Zhi Wang, 17-Oct-2025)

	Ref	Expression
Hypotheses	termcbas.c	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
	termcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	termcbasmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	termcbasmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	termcid.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion	termchomn0	⊢ ( 𝜑 → ¬ ( 𝑋 𝐻 𝑌 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		termcbas.c	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
2		termcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		termcbasmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
4		termcbasmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
5		termcid.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
6		eqid	⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
7	1	termccatd	⊢ ( 𝜑 → 𝐶 ∈ Cat )
8	2 5 6 7 3	catidcl	⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) )
9	1 2 3 4	termcbasmo	⊢ ( 𝜑 → 𝑋 = 𝑌 )
10	9	oveq2d	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑋 ) = ( 𝑋 𝐻 𝑌 ) )
11	8 10	eleqtrd	⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑌 ) )
12		n0i	⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑌 ) → ¬ ( 𝑋 𝐻 𝑌 ) = ∅ )
13	11 12	syl	⊢ ( 𝜑 → ¬ ( 𝑋 𝐻 𝑌 ) = ∅ )